Non-crystalline materials having complete photonic, electronic or phononic band gaps

ABSTRACT

The invention provides an article of manufacture, and methods of designing and making the article. The article permits or prohibits waves of energy, especially photonic/electromagnetic energy, to propagate through it, depending on the energy band gaps built into it. The structure of the article may be reduced to a pattern of points having a hyperuniform distribution. The point-pattern may exhibit a crystalline symmetry, a quasicrystalline symmetry or may be aperiodic. In some embodiments, the point pattern exhibits no long-range order. Preferably, the point-pattern is isotropic. In all embodiments, the article has a complete, TE- and TM-optimized band-gap. The extraordinary transmission phenomena found in the disordered hyperuniform photonic structures of the invention find use in optical micro-circuitry (all-optical, electronic or thermal switching of the transmission), near-field optical probing, thermophotovoltaics, and energy-efficient incandescent sources.

This application is a Continuation of, and claims priority to,co-pending U.S. patent application Ser. No. 14/953,652, filed Nov. 30,2015, which claims priority to U.S. patent application Ser. No.13/379,740, filed Feb. 15, 2012, which is a national entry ofPCT/US2010/039516, filed Jun. 22, 2010, which claims priority toProvisional Application Ser. No. 61/269,268, filed Jun. 22, 2009, thecontents of which are incorporated herein in their entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under Grant No.DMR-0606415 awarded by the National Science Foundation. The governmenthas certain rights in the invention.

FIELD

Embodiments of the invention relate to materials having photonic,electronic or phononic band gaps, methods of designing such materials,and devices that comprise them.

BACKGROUND

Crystals are materials composed of elements arranged in a repetitiveorder that confers on the overall structure a wavelike configuration.These “waves” tend to interact with wavelike excitations traversing suchstructures. For example, propagation of a vibration through such astructure, although not straightforward, can be fully and predictablydescribed (as a phonon) given sufficient knowledge of the wave-structureof the material. Photonic crystals comprise two or more periodicallyrepeating dielectric materials with which photons (i.e., electromagneticwaves, which is the more straightforward choice of name for thewave/particle duality in this context) interact according to the laws ofrefraction that apply to electromagnetic waves according to Maxwell'sequations. Photonic crystals have numerous applications as efficientradiation sources, sensors and optical computer chips.

A consequence of the repetitious structure of photonic crystals is thatlight of certain wavelengths (or “frequencies,” the reciprocal ofwavelength) traveling in certain directions and orientations (or“polarizations”) will not propagate through the crystal. That is, acrossthe entire spectrum of wavelengths, certain frequency ranges (or“bands”) cannot pass through the structure. The transmitted spectrumthus has “gaps” in it. In common parlance, the property of the structurethat gives rise to a gap in the transmitted spectrum is called a “bandgap.” Band gaps that reject a band of frequencies no matter theirdirection or polarization are called “complete band gaps.” The utilityof such band gaps, like the electronic band gaps in semiconductors, liesin their susceptibility to being breached by defects intentionallyintroduced into the structure. By violating the perfect periodicarrangement of the dielectric material-elements of the crystal, apreviously prohibited frequency band is allowed passage into thecrystal, where it may be trapped, re-directed, or otherwise altered.

The usefulness of the complete band gaps in periodic structures is,however, limited by the very periodicity on which they depend.Periodicity limits the engineer of photonic crystals to symmetries thatdo not lend themselves to dielectric materials other than those withdielectric constants that produce very high dielectric contrast. Formany applications, furthermore, the band gaps in periodic structurestend to be of limited usefulness because their anisotropy makes devicesmade with them highly direction-dependent. Periodicity also limits theengineer to a narrow choice of defects, and increases the risk ofintroducing unintended defects during fabrication. Theseperiodicity-induced limitations apply also to structures sized tocontrol the passage of phonons and electrons.

Materials are needed that relax constraints imposed by periodicity butallow the artisan to engineer complete band gaps that preferably do notdepend on direction or polarization.

SUMMARY

In various embodiments, the invention provides articles of manufacture,and methods of designing and making said articles, wherein the articlecomprises a plurality of material-elements disposed with respect to oneanother as a heterostructure preferably in a non-crystalline orquasicrystalline, hyperuniform, isotropic distribution in a condensed orsolid state. Articles that embody the invention permit excitations inthe form of waves of energy to propagate through them or, depending uponthe frequency of the wave and its direction of propagation, prohibitsuch passage by reflection or trapping. Additionally, said articles havea complete band gap, preferably TE- and TM-optimized.

In preferred embodiments, the article comprises a non-crystalline,hyperuniform heterostructure comprising a plurality of material-elementsand a complete band gap. In one embodiment, said heterostructure isderived from a hyperuniform pattern of points. In one embodiment, saidpoint-pattern comprises a plurality of points disposed in a plane. Inanother embodiment, said point-pattern comprises a plurality of pointsdisposed in a d-dimensional space to create a d-dimensionalpoint-pattern. In one embodiment, said heterostructure is a polygonalheterostructure. In one embodiment, said heterostructure is a polygonalheterostructure having an azimuthal symmetry. In another embodiment,said heterostructure is a polyhedral heterostructure. In one embodiment,said heterostructure exhibits a quasicrystalline symmetry. In oneembodiment, said heterostructure comprises material-elements arrangedwith a long-range order. In another embodiment, said heterostructurecomprises disordered material-elements. In one embodiment, saidheterostructure is translationally isotropic. In one embodiment, saidheterostructure is rotationally isotropic. In one embodiment, saidmaterial-elements of said heterostructure comprise a lattice, saidlattice comprising a plurality of polygonal cells, wherein a pluralityof intersecting lines defines said polygonal cells, said lines definecell-edges, said intersections define vertices, and each said celldefines therein a polygonal cell-space. In another embodiment, saidlattice comprises a plurality of polyhedral cells, wherein saidplurality of intersecting lines defines said polyhedral cells, whereineach said cell comprises a plurality of faces, a plurality of verticesand defines therein a polyhedral cell-space.

In one embodiment, said edges, faces and vertices have disposed thereona first material-element, and said cell-spaces are filled with a secondmaterial-element. In one embodiment, said first material-element has ahigher dielectric constant than said second material element. In apreferred embodiment, said first material-element comprises silicon, andsaid second material-element comprises air. Further, in this embodiment,said first material-element is disposed on said edges or said faces atfinite thickness and each said vertex is coincident with a centroid of acylinder having a finite thickness and a finite radius.

In one embodiment, the invention provides a method of making ahyperuniform heterostructure having a complete band gap comprising thesteps of:

-   -   i) selecting a structure factor for said heterostructure,    -   ii) constructing a box of size L, said box having a first        point-pattern of points, wherein said points are spaced apart in        a translationally disordered manner, said spaced-apart points        having an average spacing, wherein said average spacing is <<L,    -   iii) constructing a Delaunay trivalent tiling from said first        point-pattern and plotting a centroid for each said tile to        create a centroid point-pattern,    -   iv) identifying for each centroid in said centroid point-pattern        a nearest-neighbor neighborhood of said tile;    -   v) constructing a plurality of lines to connect said centroids        in each said neighborhood in such a manner that (i) said        plurality of lines defines a plurality of edges or faces having        vertices, (ii) said plurality of edges or faces defines a        super-cell having therein a cell-space, and (iii) each said        super-cell surrounds an unique point in said first        point-pattern;    -   vi) constructing a heterostructure by disposing on said edges or        faces and vertices a first material-element and filling said        cell-spaces with a second material-element, and assembling said        heterostructure from a plurality of said supercells.

In one embodiment, said first point-pattern of said method comprisesvertices of a Penrose tiling.

In a preferred embodiment, said first point-pattern of said method has aparameter χ that determines a fraction of wavenumbers k within aBrillouin zone for which the structure factor S(k) is set equal to zerosuch that, as χ increases, k_(c) increases until χ reaches a criticalvalue χc, beyond which said disordered pattern attains a long-rangetranslational order.

In one embodiment, said first material-element of said method has ahigher dielectric constant than said second material element. In apreferred embodiment, said first material-element comprises silicon, andsaid second material-element comprises air. Further, in this embodiment,said first material-element is disposed on said edges or faces at afinite thickness and each said vertex is coincident with a centroid of acylinder having a finite thickness and a finite radius.

In a preferred embodiment, said centroid point-pattern exhibits a numbervariance

N_(R) ²

−

N_(R)

²∝R^(p), within a spherical sampling window of radius R, wherein p<d.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 provides point-patterns and diffraction images of the scatteringfunction S(k) for structures having (a) an isotropic butnon-hyperuniform distribution of scattering elements, (b) an isotropic,hyperuniform but disordered distribution, and (c) an anisotropic,hyperuniform distribution with five-fold symmetry (a quasicrystallinepattern).

FIG. 2 shows a map in two dimensions of a photonic structure built upfrom a hyperuniform point-pattern.

FIG. 3 shows a map in two dimensions of a quasicrystalline photonicstructure with five-fold symmetry (left panel) and the fractional bandgaps (Δω/ω_(c)) therein (as a function of the number of scatteringelements) for TM-blocking (circles), TE-blocking (squares) and completeblocking.

FIG. 4 shows a map in two dimensions of a disordered hyperuniformphotonic structure optimized for a complete band gap (left panel) and agraph showing how the TM, TE and complete band gaps behave as a functionof χ.

FIG. 5 shows a prior art representation of an optimal photonic crystal.Upper left panel shows a tessellation in two dimensions of the cells ofthe crystal; upper right panel shows its diffraction pattern and thegraph depicts the structure of the crystal in terms of its band gaps.

FIG. 6 shows a representation (tessellation, diffraction pattern andband gap structure) of a hyperuniform structure having sufficienttranslational periodicity to confer on it the five-fold symmetry of aquasicrystal.

FIG. 7 shows a representation (tessellation, diffraction pattern andband gap structure) of a disordered hyperuniform structure optimized forχ=0.35.

FIG. 8 shows a representation (tessellation, diffraction pattern andband gap structure) of a disordered hyperuniform structure optimized forχ=0.4.

FIG. 9 shows a representation (tessellation, diffraction pattern andband gap structure) of a disordered hyperuniform structure optimized forχ=0.5.

FIG. 10 shows a tessellation map of a disordered hyperuniform pointpattern and (in FIG. 10 CONT.) a photograph of an actual heterostructurebased on the map, fabricated from a dielectric polymer.

FIG. 11 is a photographic image of a pattern made by transmittingmicrowave radiation through the heterostructure shown in FIG. 10. Theconcentric ring identified by the arrow arises in the image because of aTE photonic band gap in the structure.

DEFINITIONS

To facilitate an understanding of the various embodiments of thisinvention, a number of terms (which may be set off in quotation marks inthis Definitions section) are defined below. Terms defined herein(unless otherwise specified) have meanings as commonly understood by aperson of ordinary skill in the areas relevant to the present invention.As used in this specification and its appended claims, terms such as“a”, “an” and “the” are not intended to refer to only a singular entity,but include the general class of which a specific example may be usedfor illustration, unless the context dictates otherwise. The terminologyherein is used to describe specific embodiments of the invention, butthe usage of any particular term does not delimit the invention, exceptas outlined in the claims.

The phrase “chosen from A, B, and C” and the like, as used herein, meansselecting one or more of A, B, C.

As used herein, absent an express indication to the contrary, the term“or” when used in the expression “A or B,” where A and B refer to acomposition, product, etc., means one or the other, or both. As usedherein, the term “comprising” when placed before the recitation of stepsin a method means that the method encompasses one or more steps that areadditional to those expressly recited, and that the additional one ormore steps may be performed before, between, and/or after the recitedsteps. For example, a method comprising steps a, b, and c encompasses amethod of steps a, b, x, and c, a method of steps a, b, c, and x, aswell as a method of steps x, a, b, and c. Furthermore, the term“comprising” when placed before the recitation of steps in a method doesnot (although it may) require sequential performance of the listedsteps, unless the context clearly dictates otherwise. For example, amethod comprising steps a, b, and c encompasses, for example, a methodof performing steps in the order of a, c, and b; c, b, and a, and c, a,and b, etc.

Unless otherwise indicated, all numbers expressing quantities ofingredients, properties such as molecular weights, reaction conditions,etc., as used in the specification and claims, are to be understood asbeing modified in all instances by the term “about.” Accordingly, unlessindicated to the contrary, the numerical parameters in the specificationand claims are approximations that may vary depending upon the desiredproperties sought to be obtained by the present invention. At the veryleast, and without limiting the application of the doctrine ofequivalents to the scope of the claims, each numerical parameter shouldbe construed in light of the number of reported significant digits andby applying ordinary rounding techniques. Notwithstanding that thenumerical ranges and parameters describing the broad scope of theinvention are approximations, the numerical values in the specificexamples are reported as precisely as possible. Any numerical value,however, inherently contains standard deviations that necessarily resultfrom the errors found in the numerical value's testing measurements.

The term “not,” when preceding and made in reference to any particularnamed composition or phenomenon, means that only the particularly namedcomposition or phenomenon is excluded.

The term “altering” and grammatical equivalents as used herein inreference to the level of any substance and/or phenomenon refers to anincrease and/or decrease in the quantity of the substance and/orphenomenon, regardless of whether the quantity is determinedobjectively, and/or subjectively.

The terms “increase,” “elevate,” “raise,” and grammatical equivalentswhen used in reference to the level of a substance and/or phenomenon ina first instance relative to a second instance, mean that the quantityof the substance and/or phenomenon in the first instance is higher thanin the second instance by any amount that is statistically significantusing any art-accepted statistical method of analysis. The increase maybe determined subjectively, when a person refers to his subjectiveperception of pain, etc., for example, or objectively, when a person'sobservable behavior indicates pain. Correspondingly, the terms “reduce,”“inhibit,” “diminish,” “suppress,” “decrease,” and grammaticalequivalents when used in reference to the level of a substance and/orphenomenon in a first instance relative to a second instance, mean thatthe quantity of substance and/or phenomenon in the first instance islower than in the second instance by any amount that is statisticallysignificant using any art-accepted statistical method of analysis.

A number of the terms used herein are defined in geometry. Someembodiments of the invention, for example, relate to the “construction”of geometrical figures, whether with pencil, paper and compass or bymeans of computer programs, as distinct from the “construction” ofobjects, systems, etc. with physical materials. The term may be usedherein in both senses, and its meaning in any instance will bedetermined by the context. A constructed “line” herein may be straight,curved, continuous, discontinuous or segmented, signifyingconnectedness, direction, segregation, etc. as the context so admits.Various geometrical forms such as but not limited to “triangle,”“trihedron,” “box,” “face,” “wall,” “cell,” “locus,” “point,”“centroid,” and “cylinder” are referred to herein. In general, theseterms will be used herein within their meaning in Euclidian space or inthe Euclidian plane, but may be represented in other spaces bymathematical transformation. Thus, The term “space” herein encompassesany spaces defined in topology, regardless of dimension. In thisconnection, the term “trivalent” refers equally to planar triangles andto trihedrons.

Other geometrical references made herein include “congruence,” whichrefers to the identity (but for orientation in space) of a form withanother form; “similarity,” which refers to forms that are congruent butfor scale, and “symmetry,” which refers to a property that a geometricobject possesses if at least one translation, rotation, reflection orinversion operation can be performed on it that recapitulates theundisturbed object. The latter term is not to be confused with “order.”One can apply these tests of symmetry to disordered patterns as well asto ordered patterns. Objects for which all rotations are symmetric are“rotationally isotropic.” Objects for which all translations aresymmetric are “translationally isotropic.” “Translational invariance,”as used herein, refers to a condition that may be revealed in a testobject (i.e., an object being inspected for the condition) by moving anidentical copy of the test object on (or through) the test object alonga vector (i.e., in a defined direction), and finding that a definedpoint (or point pattern) in the copied object is congruent with aninfinite discrete set of points in the object. Note that this definitionimplies, at least with respect to the direction of the vector, an objectof infinite size (i.e., having no “edge).

The term “structure,” as used herein is not intended to be limited tothe concept of an assembly of classically mechanical elements. Some“structures” herein (e.g., the first Brillouin zone, described infra)may be better represented and analyzed in “reciprocal space” than inreal space. To distinguish topologically defined structures fromassemblies of classically mechanical elements, the latter are referredto herein as “compositions,” “articles,” or “articles of manufacture”assembled using “material-elements” made of “materials” (e.g., silicon).A given composition may comprise one or more species of material; a“material-element” does not imply that each such element is of only onesize, shape, constitution or function.

A “crystal” usually refers to a system of bonded atoms arranged inperiodically repeating sub-systems (“cells”). The term may be usedherein more broadly to refer, without limitation, to any assemblage ofpoints, lines, planes, volumes or other elements arranged in a periodicmanner Such elements may be geometrical abstractions (“points”) or theymay comprise actual physical material. Typically, in systems of two ormore dimensions, the periodically arranged elements are disposed inrelation to one another in the form of a latticework. Familiarly, the“points” comprising the latticework are occupied by atoms chemicallybonded to one another at the vertices of the faces of the lattice andthe electron “cloud” associated with each such atom accounts,collectively, for the interference with the passage of electromagneticenergy (e.g., X-rays) through the lattice. Such lattices may have two-,three- or n-dimensions, and may be treated as being infinite: that is,having indefinite boundaries that encompass an essentially infinitenumber of cells. A particularly useful primitive cell for analyticalpurposes is the so-called “Wigner-Seitz” cell, found by identifying, fora selected vertex point, the set (or “locus”) of vertex points in thelattice that are closer to the selected point than to any of the otherpoints in the lattice. These points are in the “neighborhood” or“point-neighborhood” of the selected point (triangles having a vertex ata selected point are in the selected point's “triangle neighborhood”). Aplurality of such cells can be assembled without “gaps” in between, or“overlaps,” (i.e., the cells are “tessellated”) and the result maps theoverall structure of the lattice. One can pass an electromagneticfield-wave (X-rays, for example) through a real crystal, such that thewave is diffracted by its interaction with the electron clouds in thecrystal, finally to illuminate a screen at a distance beyond. One willobserve on the screen a pattern of light and dark spots, lines and/orcircles (i.e., a “diffraction pattern”). The pattern is not an image ofthe crystal but it is the reciprocal of such an image. It is the crystallattice in “reciprocal space.” From it, one can construct a primitivecell that captures (the reciprocal of) the lattice structure of theentire crystal. This primitive cell is called the “first Brillouin zone”(or, for practical purposes, simply the “Brillouin zone”).

A “photonic crystal” affects the passage of photons (i.e.,electromagnetic waves) in particular by virtue of interfaces that occurperiodically between two or more materials comprising the crystal.Because the materials have different refractive indices, light passingfrom one material into the other at an interface between them slows down(or speeds up), which bends the path of the light. The ensemble of suchinterfaces in a photonic crystal behaves collectively as if distributedin a latticework of points that scatter light. Light of a certainwavelength (or, more precisely, within a certain range or “band” ofwavelengths) entering such a crystal in a certain direction may or maynot traverse the crystal depending upon how it is refracted. Because alight wave is an electromagnetic excitation that oscillates in a plane(see infra), its traversal also depends upon the orientation of theplane in which the wave is traveling.

Light is an electromagnetic wave of energy produced by oscillatingcharges or magnets. In a vacuum, the wave propagates at 300,000kilometers per second along a straight line called the propagationdirection. In a dielectric material, it travels at a speed reduced by afactor “n,” known as the refractive index. In a dielectricheterostructure, light moves through a heterogeneous mixture ofmaterials with different dielectric constants and, hence, differentlight speeds.

At any point along the light wave, the electric field oscillates alongan axis perpendicular to the propagation direction and a magnetic fieldpropagates along an axis perpendicular to the electric field axis andthe propagation direction. Light is called linearly polarized if theelectric field axis is oriented in the same direction all along thewave. The axis of electric field axis oscillation is called thepolarization or polarization direction. In general, light wavespropagating in a certain direction can be decomposed into a combinationof two independent polarizations, conventionally chosen to beperpendicular to one another (and the propagating direction). Lightcoming from a source can be polarized, which means a majority of lightwaves traveling in the same direction will have the same axis ofelectric field oscillation. Alternatively, it may contain an equalmixture of light with both polarizations.

In two dimensional photonic materials (or three-dimensional photonicmaterials with azimuthal symmetry), the material is used in such a waythat light propagates along the two-dimensions (or in the planeperpendicular to the azimuthal direction). The polarization direction,which must be perpendicular to the propagation direction, may be purelyin the azimuthal direction; this is called TM or TM polarization or TMpolarized light. Alternatively, the polarization direction may be in thetwo dimensional plane (and perpendicular to the propagation direction);this is called TE or TE polarized light. The light may also contain amixture of TM and TE polarized light. The same nomenclature is used torefer to two independent polarizations in three-dimensional photonicmaterials without azimuthal symmetry.

To the extent electromagnetic waves (e.g., light waves) of energypassing through a composition of matter interact with dielectricelements distributed within the composition as “obstacles” to the freepassage of the energy, only waves of certain frequencies, moving incertain directions, will in fact pass through. Others, due toreflection, refraction or diffractive interference, will not. For agiven direction of light travel, the composition may permit one or moredistinct ranges, or “bands,” of frequencies to pass through whileprohibiting electromagnetic energy in other states from completelytraversing it. These prohibited ranges define “band gaps” in thecomposition. It is not intended that the terms “band” and “band gap”herein have reference solely to electromagnetic waves. Energy, be itelectromagnetic, electronic, acoustic, or otherwise, can exist in acomposition only in certain states (frequencies, polarizations, etc.).The energy may, without limitation, be photonic (affected by electricalinsulators, i.e., dielectric “obstacles”), electronic (affected byelectrons transiently associated with atomic nuclei in a material),acoustic (affected by the mass of constrained but elastically vibratingatoms or “phonons”), or even a surface wave on a body of water (affectedby macroscopic objects in its path) (Jeong et al, Applied PhysicsLetters 85:1645-1647, 2004).

As alluded to above, photonic band gaps do not exclude all lightwaves,only those of certain wavelengths traveling in certain directions and incertain modes. A “complete band gap,” as the term is used herein, is aband gap that prohibits the passage of both TE-mode polarized andTM-mode polarized light. Complete band gaps are not necessarily equallyproficient at blocking light travel in each mode. For example, a givencomplete band gap may prohibit TB-polarized light robustly, andTM-polarized light weakly. The band gap, although complete, is not“optimized.” In fact, the excluded range of frequencies for the TE-modeis generally substantially different from the excluded range offrequencies for the TM-mode, so the bandgap is complete only where thetwo ranges intersect, i.e., in the range where both polarizations areexcluded. Preferred embodiments of the present invention permit theconstruction of optimal complete band gaps in heterostructures.

Energy is carried in an electromagnetic wave or oscillating “field” inone direction indefinitely at the speed of light (in a vacuum) until thewave encounters an object that reflects it, absorbs it, delays it (as in“refraction”), or distorts it (as in “diffraction”). Electromagneticwaves can also interact (“interfere”) with one another, additively toincrease amplitude, or subtractively to extinguish all amplitude.

The amount of electromagnetic energy that impinges on an object in anygiven period of time depends OD how many waves (counted by their“crests,” for example) reach the object during that period (“frequency”)and the height of the wave (the “amplitude’). Short wavelengths meanhigh frequency, which, for a given amplitude, means more energy. Thus,electromagnetic energy can be denominated as frequency or as the numberof waves that pass through a given space at a given speed(“wavenumber”). It is to be noted that since waves have both speed (ormagnitude) and direction, they are vectors, symbolized herein by thebolded letter k. The magnitude of k is |k|. The (angular) wavenumber (k)of an electromagnetic wave is inversely proportional to the wavelengthand is equal to |k|.

A “point pattern,” as the term is used herein, refers to a pattern ofpoints that may be used to derive a map or “blueprint” on which one mayfabricate articles that embody the invention. Such articles may befabricated to manipulate the flow of energy carried by waves, includingbut not limited to light energy. For example, the flow of energy carriedby acoustic waves can be manipulated in some embodiments of theinvention. In some embodiments, the invention provides a map having atwo dimensional point-pattern. In some embodiments, the inventionprovides a map having a three-dimensional point-pattern. Advantageously,the patterns or maps are accessible for use in a computer. In this form,the structures they represent may be unbounded (to the extent computerspermit). In one embodiment, useful in designing desired fabrications,the point-pattern is two-dimensional and has definable within it acompletely convex (typically circular) “sampling window” of radius Rthat may be varied in size. In one embodiment, the point-pattern isthree-dimensional and has definable within it a completely convex(typically spherical) window. It is not intended that embodiments belimited to two or three dimensions. One-dimensional and n-dimensionalpoint-patterns are also contemplated.

The “structure factor” of a structure relates to the “order” that thepoints of interest collectively assume in a structure under one oranother condition, in the sense that the structure factor is a measureof the probability that the structure will scatter a wave of wavenumberk, which probability, in turn, is affected by the arrangement of thescatter-points in the structure. A structure factor may also be referredto as a “structure function,” “power spectrum,” “power density spectrum”or “S(k)”. The “order” of a point-pattern in a structure relates hereinto a property exhibited by a population of points arranged ordistributed in an array along a line, or in n-dimensional space. Thatproperty may be measured in terms of “number variance.” In any samplingwindow in which one inspects a region of the structure, one will find acertain point-density (e.g., points/unit volume). By moving theinspection window repeatedly without changing its size (i.e., its volumein a three-dimensional structure), and counting the number of pointsencountered in the window each time the window is moved, one may readilydetermine the average number of points observed and the variance in thenumber observed for that window size. As one changes the size (surfaceor volume) of the window, the number variance in the observedpoint-density will vary in a way that depends upon how the points in thepopulation are ordered. If the points are distributed randomly in thestructure (i.e., according to Poisson statistics), the increase in thevolume of the observation window and the increase in number variancewill be equal. If distributed hyperuniformly, the increase in numbervariance will grow only as a fraction of the increase in the volume.

In physical systems, structure functions and power spectra provideinformation about how points of mass, charge, energy, etc. distributethemselves in the system. A crystal is an example of a physical systemcomprising points that scatter impinging radiation (“scatter-points”).The radiation that scatters from the crystal depends on the crystal'sstructure function which, in turn, depends on the extent to which—andthe manner in which—the density of scatter-points fluctuates through thecrystal.

To rationalize the concept of “order” in point-patterns, at least threetypes of “order” are referred to herein. “Random order” refers tostructures having point-patterns such that the statistics ofpoint-density fluctuations in any (normalized) sample of the structure(e.g., the sample variance) are consistent with a random (or “Poisson”)distribution. Stated quantitatively, the number variance of a purelyrandom point-pattern in two dimensions or three dimensions variesexactly as the area (d=2) or the volume (d=3) of the window varies.“Homogeneous” or “uniform” order refers to structures whosepoint-density fluctuations are statistically the same fromsample-to-sample. Stated in another way, the variance of anystatistically homogeneous, isotropic point pattern grows more slowlythan the window's volume grows, but cannot grow more slowly than thewindow's surface grows. The number variance of a hyperuniform pointpattern grows more slowly than the window's volume—by a fraction that isstrictly less than one.

However, the number variance of a major subclass of homogeneous,isotropic point patterns grows exactly as the surface grows. Thissubclass is also referred to herein as “hyperuniform” or“superhomogeneous.” This one subclass of hyperuniform patterns is called“stealthy” because, for a certain set of wavelengths (that is, where thewavenumber ranges from k=0 to k=+k_(e)), a light-scattering structurebuilt according to such a point-pattern will not scatter light: for thatset of wavelengths, the structure factor is exactly zero (S_(k)=0).Quantitatively,

N_(R) ²

−

N_(R)

²=AR^(p), where the brackets refer to averages over many independentsampling windows of radius R. The relation determines the numbervariance in an average window of radius R (N_(R) being the number ofpoints lying within the window), where p≥d−1 and p<d. This means thatthe number variance must grow more slowly than the volume of the windowin three dimensions (that is, 2<p<3), or more slowly than the surfacearea of the window in two dimensions (1<p<2). For example, it is commonfor p to be equal to 2 or 1 in three or two dimensions, respectively.The test for hyperuniformity in point-patterns is provided in greaterdetail by Torquato et al. in Phys. Rev. E 68: 41113, 2003. Theproportionality constant A for the relation above determines the degreeof hyperuniformity, smaller values reflecting greater hyperuniformity.

Described below is an “inverse” use of the “collective coordinates”method (Uche et al., Phys. Rev. E 74:031104, 2006) to design pointpatterns from whatever scattering characteristics may be desired for aparticular structure. Collective coordinates are derived frommeasurements of the co-ordinates or locations (plotted as wavevectors)of particles in their density field. In the inverse, the wavevectorsassociated with the desired scatter are employed to locate theappropriate scatter-points for the structure. Essentially, the exerciseamounts to constraining some coordinates (i.e., reducing degrees offreedom) and not others. The system quantity χ is the ratio of thenumber of constrained degrees of freedom to the total number of degreesof freedom in the system. Essentially, it equals the fraction ofwavenumbers k within the Brillouin zone that are set to zero Thepeculiar tendency of “stealthy” hyperuniform structures to show no longrange order persists as χ increases until χ reaches adimension-dependent critical value χc, beyond which the pattern attainslong-range translational order. For two-dimensional systems, χc˜0.77.

DETAILED DESCRIPTION

In one aspect, embodiments of the invention provide methods of designingstructures of use in emitting, transmitting, amplifying, detecting andmodulating energy or energy quanta. The methods apply in particular todesigning structures that emit, transmit, amplify, detect or modulatequanta of light (i.e., photons) or the equivalent thereof in the form ofelectromagnetic waves. The methods are applicable especially tostructures that can be assembled without the degree of precisionrequired by structures that rely on periodicity (e.g., crystals) toimpart their effects on photons or, more pertinently in the presentcontext, electromagnetic waves.

Structures that affect the passage of light therethrough generallycomprise elements that scatter light. These elements form a variety ofgeometric patterns that can be represented diagrammatically ascollections of points or “scatter-points.” Light that passes through agiven collection of scatter-points, whether it arises from outside thestructure or from within the structure, forms a diffraction pattern whenit emerges. The diffraction pattern is a function of the geometricscatter-point pattern, as illustrated in FIG. 1. One can see byinspection that the point pattern shown in panel A is disordered.Inspection of the point pattern in panel B also exhibits disorder, butthe impression of “clustering” is less prominent. Neighboring points maybe quite close to one another but, over the entire field of view, highlydense clusters are not discernible. The sense of randomness is muted.More formally stated, fluctuations in density grow with thecircumference or radius rather than area. This is a hallmark ofhomogeneity (uniformity).

Many hyperuniform structures show translational periodicity.Hyperuniform patterns can be translationally disordered or ordered, andisotropic or anisotropic. Regular (periodic) crystals, which areanisotropic and translationally disordered, are hyperuniform by thecriterion defined above. Generally, quasicrystals also qualify. Althoughquasicrystals are aperiodic in the sense that they lack periodictranslational symmetry in real space, their diffraction patterns (i.e.,their reciprocal lattices) are consistent with periodicity. Thediffraction patterns consist of Bragg peaks, and their number varianceand S(k) are consistent with the definition of hyperuniformity. Anexample of a quasiperiodic crystal having five-fold symmetry and ahyperuniform structure is provided in panel C of FIG. 1 and in FIG. 6.Point-patterns that conform to a Poisson (i.e., random) distribution arenot periodic and are not hyperuniform. However, some disordered (andnon-periodic) point patterns, reflected in certain embodiments of thepresent invention, are hyperuniform.

A prime objective in designing photonic crystals is to build into themcomplete photonic band gaps. The ability to manipulate the flow of lightdepends on such band gaps. It has generally been thought that theperiodic nature of crystals is a determinant of their ability toaccommodate complete photonic band gaps. Surprisingly, the applicantshave found, in simulations, that complete band gaps can be incorporatedinto some non-crystalline (i.e., non-periodic) materials, includingdisordered materials. Therefore, in another aspect, embodiments of theinvention provide articles comprising arrangements of structuralelements that need not confer periodicity on the composition to providecomplete photonic band gaps in the composition. In fact, embodiments ofthe invention include any article of manufacture that relies on acomplete photonic, phononic or electronic band gap for functionality,and is non-crystalline (i.e., non-periodic) but is“disordered/hyperuniform.” To make any such article, one proceedsaccording to the guidance that the prior art provides for making theperiodic version of the article, but simply substitutes adisordered/hyperuniform non-crystalline pattern. By way of non-limitingexample, U.S. Pat. No. 6,869,330 to Gee et al. discloses a method forfabricating a photonic crystal from tungsten for use in an incandescentlamp having improved efficiency. A mold is fashioned lithographically ina silicon substrate according to a pattern dictated by aphotolithographic etch mask. The pattern in the mask is characterized bya periodicity but, according to the instant invention, may becharacterized by the disordered/hyperuniform criterion instead, therebyproducing a non-crystalline incandescent emitter having improvedefficiency compared to conventional tungsten filaments. U.S. Pat. No.6,468,823 to Scherer et al. discloses a method by which devices thatcomprise photonic crystals (such as waveguides, microcavitics, filters,resonators, lasers, switches, and modulators) can be made using atwo-dimensional photonic crystal structure. The structure is based onpatterning by means of mask lithography. Again, by laying out thepatterning such that the disordered/hyperuniform condition is achieved,the present invention may be embodied in waveguides, microcavities,filters, resonators, lasers, switches, modulators, etc.Three-dimensional photonic crystals can also be fabricated bylithography. U.S. Pat. No. 7,588,882 to Romanato et al. is exemplary.Again, the present invention may be embodied in devices such as thosenoted in Romanato et al. by substituting a disordered/hyperuniformnon-crystalline pattern for the conventional crystalline pattern.Disordered/hyperuniform non-crystalline versions of selective band-passfilters and photovoltaic solar cells are also within the scope of theinvention and can be made for the X-ray, ultraviolet, visible, infraredand microwave electromagnetic radiation regimes using, for example, thelayer growth techniques set forth in U.S. Pat. No. 6,064,511 to Fortmannet al. In this case, hydrogen radical beams are directed onto asubstrate through a plurality of collimators laid out in a desiredpattern, which pattern may be disordered/hyperuniform.

The acoustic regime is also susceptible to manipulation indisordered/hyperuniform structures having a phononic band gap. U.S.patent Application 2009/0295505 to Mohammadi et al., describes a methodof making a so-called “phononic crystal” that prohibits passagetherethrough of wave-mechanical energy of certain wavelengths. One canuse the same method to make disordered/hyperuniform non-crystallinephononic devices that filter, confine or guide mechanical energy andhence are useful for a variety of applications including wirelesscommunications and sensing. It is necessary only to substitute thedisordered/hyperuniform pattern for the crystalline pattern used byMohammadi et al.

The selective transmission of electrons through crystalline materials(such as silicon) having electronic band gaps is the basis ofsemiconduction and countless devices based thereon. Electronic band gapscan also be fabricated in amorphous (non-crystalline) silicon,advantageously by implantating self-ions into crystalline silicon(Laaziri et al. Physical Review Letters 1999, 82:3460-3463). Thus, ifone employs this method in combination with methods that are in accordwith the instant invention, the electronic regime may also be subject tomanipulation in disordered/hyperuniform structures.

Finally, another non-limiting method of fabricating a hetero structureaccording to embodiments of the invention is described in U.S. PatentApplication Publication No. 2009/0212265 to Grier et al.

The patents and patent applications, cited above by way of example andnot of limitation, are incorporated herein in their entirety for allpurposes.

Although it is not necessary for patentability to understand how aninvention, in any embodiment, works, the applicants believe thathyperuniformity may play a role. In any case, non-periodic structureshaving complete band gaps built into them constitute a new class ofmaterials, some of which may be at least as useful as many periodicstructures: the band gaps of the non-periodic (i.e., translationallydisordered) structures can be sizeable; their accessibility is notsubject to rotational symmetry limitations (they are rotationallyisotropic); and constraints on the placement of defects to control theflow of light are relaxed. Limitations that the symmetry planes inperiodic photonic crystals impose on waveguide fabrication are reducedor removed. Attainable band gap sizes (Δω/ω_(c)) are more than about 5%,preferably more than about 10%, and more preferably more than about 20%(where Δω is the “width” of the band gap, i.e., the range of prohibitedwave frequencies, and ω_(c) is the midpoint of that range). In apreferred embodiment, the invention provides a heterostructure havinghigh dielectric contrast. In one embodiment, the heterostructurecomprises silicon and air. Non-limiting alternatives to silicon includealumina, tungsten, etc. The recognition that hyperuniform structureshave these advantages will motivate efforts to improve the degree ofhyperuniformity in such structures, which in turn will penult persons ofskill in the art to use embodiments of the invention for many purposes.

To make a disordered heterostructure with a complete band gap accordingto a preferred embodiment of the invention, one may first lay out (i.e.,“construct”) a translationally disordered hyperuniform point pattern.The “collective coordinate” protocol from Batten et al. (2008) enablesone to create a large class of tailored hyperuniform point patterns byusing a large class of targeted functional forms for the structurefactor S(k). Given a target structure factor S(k) corresponding to adesired hyperuniform point pattern, the algorithm starts with an initialarbitrary configuration of points within a simulation box. Successiveconfigurations of points are then sequentially moved according to anoptimization technique that in the final steps results in the targetedstructure factor. An example of a class of hyperuniform point patternsthat can be used to make disordered heterostructures are “stealth” pointpatterns. Such patterns have a structure factor S(k) that is preciselyequal to zero for all |k|<k_(c) for a selected (positive) value of thecritical wavenumber k_(c) and tends to unity for large k values. Suchstructures are referred to as “stealthy” because they completelysuppress scattering for |k|<k_(c). The structure is therefore invisibleat wavelengths that meet this criterion. Provided that k_(c) is abovesome threshold value, the disordered heterostructure derived from thesestealthy patterns will have a complete band gap. By tuning k_(c), onecan increase the size of the band gap to sizes comparable to somephotonic crystals.

In further detail, the method utilizes an inverse approach: oneprescribes scattering characteristics (e.g., absolute transparency) andconstructs many-body configurations that give rise to these targetedcharacteristics. One first applies the methodology initially forstructureless (i.e., point) particles and then generalizes to structuredparticles, colloids, etc.

It is to be noted that systematically increasing the system size has noeffect on the degree of disorder. Constructed configurations remaindisordered in the infinite-volume limit (i.e., where inter-particledistances are much less than the size of the system).

“Stealth” materials refer to many-particle configurations thatcompletely suppress scattering of incident radiation for a set of wavevectors, and thus, are transparent at these wavelengths. Periodic (i.e.,crystalline) configurations are, by definition, “stealthy” since theysuppress scattering for all wavelengths except those associated withBragg scattering. The method discussed here, however, constructsdisordered stealth configurations that prevent scattering only atprescribed wavelengths with no restrictions on any other wavelengths.

To characterize the local order of an ensemble, one uses pairinformation in real space via the pair correlation function g₂(r) and inreciprocal space through the structure factor S(k) as these functionsare experimentally accessible and used widely in many-body theories.

The pair correlation function is the normalized two-particle probabilitydensity function ρ₂(r) and is proportional to the probability ofobserving a particle center at r relative to a particle at the origin.For a statistically homogeneous and isotropic medium, the paircorrelation function depends only on the magnitude of r≡|r|, and iscommonly referred to as the radial distribution function g₂(r).

The structure factor is proportional to the intensity of scattering ofincident radiation from a configuration of N particles and is defined as

$\begin{matrix}{{S(k)} = \frac{\rho(k)}{N}} & ({M1})\end{matrix}$where ρ(k) are the collective coordinates and k are the wave vectorsassociated with the system volume and boundary conditions. Collectivecoordinates are the Fourier coefficients in the expansion of the densityfield:

$\begin{matrix}{{\rho(k)} = {\sum\limits_{i}^{N}e^{i\;{k \cdot r_{j}}}}} & ({M2})\end{matrix}$where r_(j) denotes the location of particle j. When S(k) depends onlyon the magnitude of k≡|k|, the structure factor S(k) is related to theFourier transformation of g₂(r)−1, ignoring the forward scatteringassociated with k=0,S(k)=1+∫e ^(ik·r) ^(j) [g ₂(r)−1]dr  (M3)where ρ is the number density. For highly ordered systems, both g₂(r)and S(k) contain a series of δ-functions or peaks at large r and k,indicating strong correlations at the associated pair distance. Inconfigurations without long-range order, both g₂(r) and S(k) approachunity at large r and k.

The method has the advantage of targeting pair information in reciprocalspace to construct configurations whose structure factor exactly matchesthe candidate structure factor for a set of wavelengths. In addition,the procedure guarantees that the resulting configuration is aground-state structure for a class of potential functions.

The numerical optimization procedure follows that of Uche, Stillingerand Torquato, Phys. Rev. E 74, 031104 (2006) used to tailor the small kbehavior of the structure factor. The structure factor S(k) andcollective coordinates ρ(k), defined in Eqs. M1 and M2, are related tothe quantity C(k),

$\begin{matrix}{{S(k)} = {1 + {\frac{2}{N}{C(k)}}}} & ({M4})\end{matrix}$where

$\begin{matrix}{{C(k)} = {\sum\limits_{j = 1}^{N - 1}{\sum\limits_{i = {j + 1}}^{N}{{\cos\left\lbrack {k \cdot \left( {r_{j} - r_{i}} \right)} \right\rbrack}.}}}} & ({M5})\end{matrix}$

For a system interacting via a pair potential v(r_(i)−r_(i)), the totalpotential energy can be written in terms of C(k),

$\begin{matrix}{\Psi = {{\sum\limits_{i}{\sum\limits_{j}{v\left( {r_{i} - r_{j}} \right)}}} = {\Omega^{- 1}{\sum\limits_{k}{{V(k)}{C(k)}}}}}} & ({M6})\end{matrix}$where Ω and V(k) is the Fourier transform of the pair potential functionV(k)=∫Ωdr v(r)e ^(ik·r)  (M7)

For a region of space with dimensions L_(x); L_(x)×L_(y), orL_(x)×L_(y)×L_(z) in one, two, or three dimensions, subject to periodicboundary conditions, the infinite set of corresponding wave vectors hascomponents

$\begin{matrix}{k_{\gamma} = \frac{2\pi\; n_{\gamma}}{L_{\gamma}}} & ({M8})\end{matrix}$where n_(γ) are positive or negative integers, or zero and γ=x,y,z asneeded. For example, in three dimensions, the set of wave vectors are

$\begin{matrix}{k = \left( {\frac{2\pi\; n_{x}}{L_{\gamma}},\frac{2\pi\; n_{y}}{L_{\gamma}},\frac{2\pi\; n_{z}}{L_{\gamma}}} \right)} & ({M9})\end{matrix}$

It is clear that for any positive V(k) for |k|<K but zero otherwise, theglobal minimum of the total potential energy in M6 is achieved bydriving C(k) or S(k) to its minimum value for all |k|(Uche et al., Phys.Rev. E 74: 31104, 2006).

For simplicity, one may utilize a “square mound” V(k), i.e., a functionthat is a positive constant V₀ for all k∈Q, where Q is the set of wavevectors such that 0<|k|<K, and zero for all other k. In theinfinite-volume limit, this corresponds to a system of particlesinteracting via a real-space pair potential function that is bounded,damped, and oscillating about zero at large r. (Fan et al., Phys. Rev. A44: 2394, 1991; Uche et al., Phys. Rev. E 70: 46122, 2004; Uche et al.,Phys. Rev. E 74: 31104, 2006). This choice of pair potential serves ourimmediate purposes to generate many-particle configurations with tunedscattering characteristics as a numerical tool only. Such softpotentials are of physical importance in soft-matter physics and areeasier to treat theoretically (Torquato and Stillinger, Phy. Rev. Lett.100: 20602, 2008). The specific form of V(k) is largely irrelevant inthe design of scattering patterns so long as it is positive, bounded,and has compact support up to |k|=K. To approximate physical systems,V(k) may be chosen so that there is strong repulsion in v(r), for smallr. For a cutoff radius K, there are 2M(K) wave vectors in the set Qwhere M(K) is the number of independently constrained collectivecoordinates. That is, constraining C(k) implicitly constrains C(−k) dueto the relation:C(k)=C(−k).

For a system of N particles in d dimensions, there are Nd total degreesof freedom. We introduce the dimensionless parameter χ to convenientlyrepresent the ratio of the number of constrained degrees of freedomrelative to the total number of degrees of freedom

$\begin{matrix}{\chi = {\frac{M(K)}{Nd}.}} & ({M10})\end{matrix}$

The global minimum of the potential energy defined in Eq. M6 has thevalue of

${\min_{r_{i}\mspace{14mu}\ldots\mspace{20mu} r_{N}}(\Psi)} = {{- \frac{N}{2}}{\sum\limits_{k \in Q}V_{0}}}$if and only if there exist particle configurations that satisfy all ofthe imposed constraints, which necessarily occurs for χ<1. MinimizingEq. M6 to its global minimum, for χ<1, yields ground-stateconfigurations that are stealthy for all k∈Q.

To target a specific form of the structure factor to certain nonzerovalues, such as S(k)=1, we introduce a second nonnegative objectivefunction,

$\begin{matrix}{\Psi = {\sum\limits_{k \in Q}{V\;{(k)\left\lbrack {{C(k)} - {C_{0}(k)}} \right\rbrack}^{2}}}} & ({M11})\end{matrix}$where C₀(k) is associated with the target structure factor by Eq. M4. IfEq. M11 is taken to be the potential energy of an N-body system, thentwo-, three-, and four-body interactions are present (Uche et al., Phys.Rev. E 74: 31104, 2006). Equation M11 has a global minimum of zero, forχ<1, if and only if there exist configurations that satisfy all of theimposed constraints. Minimizing Eq. M11 is used to construct super-idealgases and equi-luminous materials as ground-state configurations.

Three algorithms have been employed previously for minimizing Eqs. M6and M11: steepest descent (Fan et al., Phys. Rev. A 44: 2394, 1991),conjugate gradient (Uche et al., Phys. Rev. E 70: 46122, 2004), andMINOP (Uche et al., Phys. Rev. E 74: 31104, 2006). Steepest descent andconjugate gradient methods are line search methods that differ only intheir choice of search directions (Press, Numerical Recipes in C: TheArt of Scientific Computing, Cambridge Univ. Press, 1992). The MINOPalgorithm is a trust-region method. When far from the solution, theprogram chooses a gradient direction, but when close to the solution, itchooses a quasi-Newton direction (Dennis and Mei, J. Optim. Theory Appl.28: 453, 1979; Kaufman, SIAM J. Optim. 10: 56, 1999). Upon eachiteration, the program makes an appropriate update to approximate theHessian (Kaufman, SIAM J. Optim. 10: 56, 1999).

Neither the conjugate gradient method nor MINOP algorithm significantlybiases any subset of ground-state configurations. The resultingconfigurations are visually similar, and the ensemble-averaged radialdistribution function and structure factor produced by both methods havesimilar features. The MINOP algorithm may be advantageous because it hasbeen demonstrated to be better suited to the collective coordinateprocedure than the conjugate gradient method (Uche et al., Phys. Rev. E74: 31104, 2006).

Three sets of initial conditions were considered: random placement ofparticles (Poisson distributions), random sequential addition (RSA), andperturbed lattices (integer, triangular, and face centered cubic in one,two, and three dimensions respectively). For an RSA process, particlesare assigned a diameter and randomly and irreversibly placed in spacesuch that particles are not overlapping (Torquato, Random HeterogeneousMaterials: Microstructure and Macroscopic Properties, Springer Verlag,N.Y., 2002). At sufficiently high χ, usually χ>0.6, the constructedground-state systems apparently lose all memory of their initialconfigurations. The analyses presented in the following sections will bethose of random initial conditions. In some cases at large χ<1, a globalminimum is not found. For the results discussed here, Eqs. M6 and M11were minimized to within 10⁻¹⁷ of their respective minimum value. Allother trials were excluded from the analysis.

The region of space occupied by the N particles was limited to a line inone dimension, a square in two dimensions, and a cube in threedimensions, with periodic boundary conditions. For stealth materials,particular attention was paid to the choice of N for two and threedimensions. Minimizing Eq. M6 for large χ<1 is known to yieldcrystalline ground states (Fan et al., Phys. Rev. A 44: 2394, 1991; Ucheet al., Phys. Rev. E 70: 46122, 2004; Uche et al., Phys. Rev. E 74:31104, 2006). We choose to be consistent with previous studies. In twodimensions, N was chosen as a product of the integers 2pq, and p/q is arational approximation to 3^^(1/2) so that all particles could be placedin a triangular lattice configuration without substantial deformation.In three dimensions, N was usually chosen so that N=4s³, where s is aninteger, so that the particles could be placed in a face centered cubiclattice without deformation. In minimizing Eq. M11, N occasionally wasassigned other values.

Designing the Band Gap.

A photonic band gap structure may be computed for “stealthy” structures,including but not limited to non-crystalline structures, as follows:

After selecting and plotting a point pattern with the desired rotationalsymmetry and translational order, the next step is to determine thearrangement of dielectric materials around the selected point patternthat will produce the largest complete band gap. The applicants presentherein a novel method for transforming the selected point pattern,whether crystal, quasicrystal or disordered hyperuniform, into atessellation of cells with the selected translational and rotationalsymmetry but having a nearly optimal photonic band gap structure (withband gaps within a percent or less of absolute optimal). Because theprocedure requires varying over only two degrees of freedom, theprotocol uses much less computational resources than other methods. Inone embodiment, the invention provides a method, beginning from thedisordered hyperuniform point pattern of open circles shown in FIG. 2,as discussed below.

If the goal were simply to have a band gap that only blocks TMpolarization (electric field oscillating along the azimuthal direction,that is, perpendicular to the plane of propagation and at the same timeperpendicular to the to the TE plane), it would suffice to replace eachpoint with a circular cylinder and vary the radius of the cylindersuntil the structure exhibits a maximum TM band gap. However, this designis poor for blocking TE polarization (electric field oriented in theplane). To obtain a design that optimally blocks TE modes well, one mayconstruct a Delaunay tiling from the original two-dimensional pointpattern, find the centroid of each tile (alternatives to centroids arealso possible, and connect the centroids that surround each point totransform the centroid point-pattern into a tessellation of cells. Onemay then decorate the cell edges with walls (along the azimuthaldirection) of dielectric material of uniform width w and vary the widthof the walls until the maximal TE band gap is obtained. Finally, toobtain designs exhibiting overlapping TM and TE gaps, one may constructan optimal compromise structure by adding one other step: the verticesof the trihedral network of cell walls may be decorated with circularcylinders (black circles in FIG. 2) of radius r. For a given set ofdielectric materials, the optimal complete band gap may be achieved byvarying the only two free parameters, the cell wall thickness w and thecircular cylinder radius r. Moreover, the optimized band gap isequivalent to the fundamental band gap in periodic systems, n_(B)=N,where n_(B) is the band gap number and N is the number of points perunit cell.

Previous attempts to create band gaps in photonic quasicrystals anddisordered systems have been reported that produce either TM band gapsor TE band gaps, but not both. The focus has been in finding the widestpossible TM or TE band gap, but, as it turns out, the optimal design forone is a poor design for the other and vice versa. By contrast, theprocedure described above, based on hyperuniformity, leads to an optimalcompromise aimed at finding the widest band gap for both polarizations.The disordered/hyperuniform photonic structure shown in FIG. 4 isexemplary. Complete band gaps (diamonds) open up at χ>0.3, at whichpoint both TE (circles) and TM (squares) band gaps are sizeable.

To illustrate one embodiment (but not intending any limitation), one mayassume the photonic materials are composed of silicon (with dielectricconstant ε=11.56) and air. For disordered photonic structures, one mayconstruct a sequence of periodic approximants (i.e., disorderedstructures treated in the same way a perfectly periodic structure wouldbe treated) with a “stealthy” disordered arrangement of points asdiscussed above within a square array of length L in which the number ofpoints per unit cell N ranges from about 100 to about 500 (see thehorizontal axis of the graph on the right-hand side of FIG. 3). Forfixed χ, the simulated gap width remains essentially constant as Nvaries between 100 and 500 (FIG. 3). One may conveniently use a lengthscale a=L/√n, such that all patterns have the same point density 1/a². Asignificant band gap begins to open for sufficiently large χ˜0.35 (butwell below χ_(c)), at a value where there emerges a finite exclusionzone between neighbouring points in the real space hyperuniform pattern(FIGS. 4 and 7). In reciprocal space, this value of χ corresponds to theemergence of a range of “forbidden” scattering, S(k)=0 for k<k_(c) forsome positive k_(c), surrounded by a circular shell just beyond k=k_(c)with increased scattering. The structures built around hyperuniformpatterns with χ=0.5 are found to exhibit remarkably large TM (of 36.5%)and TE (of 29.6%) photonic band gaps making them competitive with manyof their periodic and quasiperiodic counterparts. More importantly,there are complete photonic band gaps of appreciable magnitude reachingvalues of about 10% for χ=0.5, independent of any finite L. FIG. 8 showsthe optimal photonic band gaps for the case χ=0.4, FIG. 9 for the caseχ=0.5.

For comparison, FIG. 5 shows the “layout” of an optimal photonic crystalstructure (a), its simulated diffraction pattern (b), and its simulatedphotonic band gap structure (c). The band gap structure plot representshow the energy (or frequency) of a photonic state (ω) varies withincident angle (the x-axis). Instead of showing angle explicitly, theconvention of indicating how close the incident angle is to one symmetryaxis or another is followed; the symmetry axes are labelled Γ, K, M, andthen another Γ symmetry axis. For a square lattice, for example, thiswould show how the energy varies depending on how close the incidentangle comes to hitting an edge or corner directly. The bottom curveshows, for example, that the energy varies substantially. Note that,above the 0.5 level on the y-axis, the curves “crisscross” so that onecannot draw a line from left to right without hitting a line. This means“NO BAND GAP.” That is, light impinging with this energy, at some angleof incidence, will pass through the lattice. On the other hand, notethat a range exists between the bottom curve and the next ones above itthat will accept lines drawn left to right. At some angles of incidence,the gap is wide, at others narrower. That is, the crystal structure is“anisotropic.” The quasicrystalline structure of FIG. 6 and thedisordered hyperuniform structures of FIGS. 7-9 are much more“isotropic.” Whether anisotropic or isotropic, preferred structures formost applications have relatively narrow band gaps.

As an exemplary quasicrystalline embodiment, one may obtain a sequenceof periodic approximants of a Penrose tiling pattern (or other purelyrandom structure) of size L by replacing the golden mean in expressionsfor the lattice vectors and dot products by a ratio of integers,ρ=F_(n+1)/Fn=(1/1, 2/1, 3/2, 5/3), where F_(n) is the nth Fibonaccinumber. The unit cell in this non-limiting case is rectangle-shaped andits area grows as n increases and the rational approximant approachesr=(√5+1)/2. In tests for convergence as the approximant n increases, theoptimal Penrose pattern for TM radiation alone is achieved by placing ateach vertex a dielectric disk of radius r/a=0.177 (with a equal to therhombus edge length of the underlying Penrose tiling). The TM band gapis Δω/ω_(c)=37.6% (where Δω is the gap width and ω_(c) is the value ofthe frequency at the midpoint of the gap), consistent with what has beenpreviously reported (Rechtsman et al., Phys. Rev. Lett. 101: 73902,2008). Next, one may construct a tessellation whose edges define astructure that produces a nearly optimal TE band gap. The optimalstructure in this embodiment has dielectric material of width w/a=0.103along each edge; the TE band gap is Δω/ω_(c)=42.3%. Although notintended to be limiting, the calculated TE band gap in this example isthe largest ever reported for a quasicrystal lattice. Finally, for thecomplete band gap, the optimal structure is achieved by placing alongeach edge of trihedral intersection of the network a circular cylinderof radius r/a=0.157 and setting the cell wall thickness to w/a=0.042.The band gap results shown in FIG. 6 were obtained by means of asupercell approximation and a plane-wave expansion formalism.Preferably, the process of finding the optimum complete band gapincludes convergence tests for dependence on system size, as illustratedby the plot in FIG. 6. Although not intended to be limiting, theresulting structure in this example displays a calculated complete (TMand TE) photonic band gap of 16.5%—the first complete band gap everreported for a photonic quasicrystal with five fold symmetry andcomparable to the largest band gap (20%) found for photonic crystalswith the same dielectric contrast.

Although photonic crystals have larger complete band gaps (FIG. 5),quasicrystalline and disordered hyperuniform PBG materials offeradvantages for many applications. First, both are significantly moreisotropic, which is advantageous for use as highly-efficient isotropicthermal radiation sources and waveguides with arbitrary bending angle.Second, the properties of defects and channels useful for controllingthe flow of light are different for crystal, quasicrystal and disorderedstructures. Quasicrystals, like crystals, have a unique, reproducibleband structure; by contrast, the band gaps for the disordered structureshave some modest random variation for different point distributions. Onthe other hand, due to their compatibility with general boundaryconstraints, photonic band gap structures constructed around disorderedhyperuniform patterns can provide a flexible optical insulator platformfor planar optical circuits. Moreover, eventual fabrication flaws thatcould seriously degrade the optical characteristics of photonic crystalsand perhaps quasicrystals are likely to have less effect on disorderedhyperuniform structures, therefore relaxing the fabrication constraints.

EXPERIMENTAL

In FIG. 10, a photograph of a physical realization of a hyperuniformdisordered photonic structure is presented (FIG. 10 CONT.). It wasconstructed from the tessellated point-pattern shown first in FIG. 10.The point-pattern and the layout of the cylinders in the tessellation ofthe point-pattern was developed according to the guidance providedhereinabove. The hyperuniform disordered photonic structure wasfabricated by programming the co-ordinates for the cylinders in thepattern on the left into a rapid-prototyping stereolithography device(SLA-7000™, 3D Systems®, Rock Hill, S.C.) that produces a solid plasticmodel by ultraviolet laser polymerization. Accura 25™ (3D Systems®) wasused as the pre-polymer. Programming and operation of the device wasconducted according to the manufacturer's instructions. The Viper si2™,used with WaterClear Ultra 10122™ (3D Systems®) may also be used. Thefabricated heterostructure is macroscopic, but the design can beminiaturized, e.g., for nanoparticles or other particles. For example,laser tweezers can be used for particle trapping or two-photonpolymerization can be used. Such methods allow construction of a matrixof dielectric components with a photonic bandgap in the visible.

Transmission measurements for the heterostructure of FIG. 10 can be madewith a HP Model 8510C Vector Analyzer in three bands, from 8 to 15, from15 to 26 and from 26 to 42 GHz. To approximate plane waves, a singleTE₁₀ mode is coupled through two sets of horn-attached waveguides withtwo custom-made polystyrene microwave lenses. Before the sample is putin, the transmission spectrum of the setup is recorded fornormalization.

An image formed by microwaves transmitted through the structure in thephotograph in FIG. 10 is reproduced in FIG. 11. The arrow-pointidentifies a TE band gap in the structure. Materials that confer higherdielectric contrast on the structure than the polymer material employedhere are used to open a complete band-gap in the structure. As notedabove, to accommodate a different electromagnetic regime (e.g., visiblelight), the structure is scaled down.

We claim:
 1. A method of making a design that optimally blocks TEpolarization, comprising: a) providing a two-dimensional point patternin a hyperuniform, translationally disordered manner; b) constructing aDelaunay tiling from the said two-dimensional point pattern, said tilingcomprising tiles; c) determining the centroid of each tile of saidtiling; d) connecting the centroids that surround each point totransform the centroid point-pattern into a tessellation of cells, saidcells comprising edges; e) decorating said cell edges with walls ofdielectric material; and f) varying the width of the walls until themaximal TE band gap is obtained, wherein the design comprises anon-crystalline heterostructure comprising a plurality of disorderedmaterial-elements disposed hyperuniformly therein, without periodic orquasiperiodic translational order, in a condensed or solid state, saidnon-crystalline heterostructure being isotropic and having a completeband gap.
 2. The method of claim 1, said walls comprise a trihedralnetwork having vertices.
 3. The method of claim 2, further comprisingdecorating said vertices of said trihedral network of said cell wallswith circular cylinders, said cylinders having a radius.
 4. The methodof claim 3, further comprising varying the radius of said cylinders toachieve an optimal band gap.